Defining polynomial
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\(x^{9} + 18 x^{8} + 6 x^{6} + 18 x^{5} + 21\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $9$ |
| Ramification exponent $e$: | $9$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $22$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $3$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[2, 3]$ |
Intermediate fields
| 3.3.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{3}$ |
| Relative Eisenstein polynomial: |
\( x^{9} + 18 x^{8} + 6 x^{6} + 18 x^{5} + 21 \)
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Ramification polygon
| Residual polynomials: | $2z^{2} + 1$,$z^{6} + 2$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[14, 6, 0]$ |